On the Polytope Escape Problem for Continuous Linear Dynamical Systems

The Polyhedral Escape Problem for continuous linear dynamical systems consists of deciding, given an affine function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ and a convex polyhedron $\mathcal{P} \subseteq \mathbb{R}^{d}$, whether, for some initial point $\boldsymbol{x}_{0}$ in $\mathcal{P}$, the trajectory of the unique solution to the differential equation $\dot{\boldsymbol{x}}(t)=f(\boldsymbol{x}(t))$, $\boldsymbol{x}(0)=\boldsymbol{x}_{0}$, is entirely contained in $\mathcal{P}$. We show that this problem is decidable, by reducing it in polynomial time to the decision version of linear programming with real algebraic coefficients, thus placing it in $\exists \mathbb{R}$, which lies between NP and PSPACE. Our algorithm makes use of spectral techniques and relies among others on tools from Diophantine approximation.Comment: Accepted to HSCC 201

The Algebraic Approach to Phase Retrieval and Explicit Inversion at the Identifiability Threshold

We study phase retrieval from magnitude measurements of an unknown signal as an algebraic estimation problem. Indeed, phase retrieval from rank-one and more general linear measurements can be treated in an algebraic way. It is verified that a certain number of generic rank-one or generic linear measurements are sufficient to enable signal reconstruction for generic signals, and slightly more generic measurements yield reconstructability for all signals. Our results solve a few open problems stated in the recent literature. Furthermore, we show how the algebraic estimation problem can be solved by a closed-form algebraic estimation technique, termed ideal regression, providing non-asymptotic success guarantees

On the number of types in sparse graphs

We prove that for every class of graphs $\mathcal{C}$ which is nowhere dense, as defined by Nesetril and Ossona de Mendez, and for every first order formula $\phi(\bar x,\bar y)$, whenever one draws a graph $G\in \mathcal{C}$ and a subset of its nodes $A$, the number of subsets of $A^{|\bar y|}$ which are of the form $\{\bar v\in A^{|\bar y|}\, \colon\, G\models\phi(\bar u,\bar v)\}$ for some valuation $\bar u$ of $\bar x$ in $G$ is bounded by $\mathcal{O}(|A|^{|\bar x|+\epsilon})$, for every $\epsilon>0$. This provides optimal bounds on the VC-density of first-order definable set systems in nowhere dense graph classes. We also give two new proofs of upper bounds on quantities in nowhere dense classes which are relevant for their logical treatment. Firstly, we provide a new proof of the fact that nowhere dense classes are uniformly quasi-wide, implying explicit, polynomial upper bounds on the functions relating the two notions. Secondly, we give a new combinatorial proof of the result of Adler and Adler stating that every nowhere dense class of graphs is stable. In contrast to the previous proofs of the above results, our proofs are completely finitistic and constructive, and yield explicit and computable upper bounds on quantities related to uniform quasi-wideness (margins) and stability (ladder indices)

Reasoning about Measures of Unmeasurable Sets

International audienceIn a variety of reasoning tasks, one estimates the likelihood of events by means of volumes of sets they define. Such sets need to be measurable, which is usually achieved by putting bounds, sometimes ad hoc, on them. We address the question how unbounded or unmeasurable sets can be measured nonetheless. Intuitively, we want to know how likely a randomly chosen point is to be in a given set, even in the absence of a uniform distribution over the entire space. To address this, we follow a recently proposed approach of taking intersection of a set with balls of increasing radius, and defining the measure by means of the asymptotic behavior of the proportion of such balls taken by the set. We show that this approach works for every set definable in first-order logic with the usual arithmetic over the reals (addition, multiplication, exponentiation, etc.), and every uniform measure over the space, of which the usual Lebesgue measure (area, volume, etc.) is an example. In fact we establish a correspondence between the good asymptotic behavior and the finiteness of the VC dimension of definable families of sets. Towards computing the measure thus defined, we show how to avoid the asymptotics and characterize it via a specific subset of the unit sphere. Using definability of this set, and known techniques for sampling from the unit sphere, we give two algorithms for estimating our measure of unbounded unmeasurable sets, with deterministic and probabilistic guarantees, the latter being more efficient. Finally we show that a discrete analog of this measure exists and is similarly well-behaved